how to Rig / Animate a Multi Lobe Camshaft
Have you considered using  Shrinkwrap constraints?
Here. the valve-rod is parented to an Empty, which is constrained to be projected onto the surface of the cam, down its own Y axis..

If the slight penetration is a bother, maybe that could be tweaked out by keyframing or driving the constraint's offset, or influence? Or by having a hidden, slightly reshaped cam, to displace the Empty,avoiding intermittent intersection with the rod.
Example settings.. the Empty is constrained to the Cam, projected.